SPECIAL ALGEBRAIC TRANSFORMATIONS 



217 



and consequently, by inversion, for the primitive n^^ root of w^ 2= 



vw. If therefore in the comjpound linhage jiist described z repre- 

 sents this root, then the points u, u^, . . . , u^^^ of the closed link- 

 age will represent the other roots. 



4. Linkage for Special Collineations. 



The linkage of Fig. 2 may be slightly generalized and then com- 

 bined with a linkage of the same kind, as shown in Fig. 6. Take 

 two rhombs AEPF and ACBD with the common joint, or pivot A; 

 join E and C, and F and D by two equal links EC = FD so that 



TT 



<EAC=FAD^ — From plane geometry it follows easily <EAF = 



<CAD, i. e., that the two rhombs are similar; further that PA±AB, 

 no matter how the linkage may be distorted. This linkage realises 



Fig. 6 



therefore a variable right triangle PAB whose angles are constant. 

 Joining in a symmetrical manner the rhomb BI1P'G = AEPF to the 

 previous linkage (CG = CE, HD=FD), a variable rectangle ABPP' 

 is obtained whose sides have a constant 7'atio, Fig. 6. 

 3 



